(************** Content-type: application/mathematica **************

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Notebook[{

Cell[CellGroupData[{
Cell["Cassinian Oval", "Title"],

Cell[CellGroupData[{

Cell["Note", "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[TextData[{
  "This Mathematica notebook is from the WWW project A Visual Dictionary of \
Special Plane Curves. Copyright \[Copyright] 1995-2002 by Xah Lee. (",
  ButtonBox["http://xahlee.org/",
    ButtonData:>{
      URL[ "http://xahlee.org/"], None},
    ButtonStyle->"Hyperlink"],
  ")"
}], "Text",
  Evaluatable->False,
  AspectRatioFixed->True]
}, Open  ]],

Cell[CellGroupData[{

Cell["Graphics illustrating Cassinian Oval as a torus section", "Subtitle"],

Cell[CellGroupData[{

Cell["Packages and Initializations", "Subsubsection",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell[BoxData[{
    \(Needs["\<Graphics`ImplicitPlot`\>"]\), "\n", 
    \( (*MathSource\ \(Item : 0200 - 529\)*) \)}], "Input"],

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    InitializationCell->True,
    AspectRatioFixed->True,
    FontColor->RGBColor[0.439445, 0.0170748, 0.0170748]]
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Cell[BoxData[{\(Needs["\<ExtendGraphics`Geometry3D`\>"]\), "\
\[IndentingNewLine]", 
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      RowBox[{"(*", 
        RowBox[{"MathSource", " ", 
          RowBox[{"Item", ":", 
            ButtonBox[\(0208 - 976\),
              ButtonData:>{
                URL[ 
                "http://www.mathsource.com/Content/Enhancements/Graphics/3D/\
0208-976"], None},
              ButtonStyle->"Hyperlink"]}]}], "*)"}]}]}], "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell[BoxData[
    \(\($HistoryLength = 5;\)\)], "Input"],

Cell["\<\
Clear[ Slice3DGraphics]

(*Clip3D is a function in Tom W.J.'s ExtendGraphics`Geometry3D` Package*)
(*It is not well documented. Here's a replacement usage documentation*)
Clip3D::usage= \"Clip3D[ gPrim, Plane[p,v] ]
\twill take a list of Graphics3D primitives gPrim, and return a list of
\tGraphics3D primitives that is on one side of the plane Plane[p,v],
\twhere v is the normal vector and p is a point on the plane. Primitives
\tthat contain points on both sides are cut. Example:
\tNeeds[\\\"Graphics`Shapes`\\\"];
\tShow@ Graphics3D@ Clip3D[ Torus[2,1,15,15], Plane[{2,0,0}, {-1,0,0}]]\";

Slice3DGraphics::usage = \"
\tSlice3DGraphics[ graphics3DPrimitives,  sliceAxis, {p1, p2,...,pn} ]
\twill return a list of Graphics3D primitives that is parts of the
\tgiven graphics3DPrimitives sliced along planes passing through
\ta list of nPoints {p1, p2,...,pn} and normal to sliceAxis vector.
\tGraphics that lies outside the first and last cutting plane will
\tbe clipped.\tThe number of Graphics3D objects returned depends on
\tpositions of nPoints.
\tThe nPoints that define the slicing plane must be
\tin order; towards the direction of sliceAxis vector.
\tsliceAxis is a vector of the form {x,y,z}.
\tSlice3DGraphics calls functions in the package \
ExtendGraphics`Geometry3D.\";

Slice3DGraphics[ gp_List, axis_List, throughPoints_List ]:=
Module[{pieces, remain},
\tremain = gp;
\tpieces = Table[Null, {Length@ throughPoints-1} ];
\tDo[
\t\tremain = Clip3D[ remain, Plane[ throughPoints[[i]], axis] ];
\t\tpieces[[i]] = Clip3D[ remain, Plane[ throughPoints[[i+1]], -axis] ],
\t\t{i, Length@ pieces}
\t];
\tpieces
]\
\>", "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell["\<\
torusSweepingCircles::usage=\"
torusSweepingCircles[b,a,n,{tMin, tMax},{thetaMin,thetaMax}]
will return Graphics3D primitives of circular arcs that represents
the path of surface of revolution for a torus. Example:
Show@ Graphics3D@ torusSweepingCircles[4,1,6,{1,5},{0, Pi}]\";

Clear[torusSweepingCircles]

torusSweepingCircles[b_,a_,n_,{tMin_, tMax_},{thetaMin_,thetaMax_}]:=
\tFirst@ ParametricPlot3D[
\t\t{ (b+First@#) Cos[t], (b+First@#) Sin[t], Last@ #},
\t\t{t, tMin, tMax},
\t\tDisplayFunction->Identity
\t]& /@
\t\t( (a{Cos[#], Sin[#]})& /@
\t\t\tRange[thetaMin,thetaMax,(thetaMax-thetaMin)/(n-1)]
\t\t)\
\>", "Input",
  InitializationCell->True,
  AspectRatioFixed->True],

Cell["\<\
Clear[torusGeneratingCircles]

torusGeneratingCircles::usage=\"
torusGeneratingCircles[b,a, n, {tMin,tMax}, {thetaMin,thetaMax}]
will return Graphics3D primitives that represents the generating
circles of a torus. Example:
Show@ Graphics3D@ torusGeneratingCircles[3,1, 5, {1,5}, {0,6}]\";

torusGeneratingCircles[b_,a_,n_,{tMin_,tMax_},{thetaMin_,thetaMax_}]:=
\tFirst@ ParametricPlot3D[#, {theta, thetaMin, thetaMax},
\t\tDisplayFunction->Identity]& /@
\t\t(Rotate3D[{a Cos[theta],0,a Sin[theta]}+{b,0,0}, -#,0,0]& /@
\t\t\tRange[tMin, tMax, (tMax-tMin)/(n-1)]
\t\t)\
\>", "Input",
  InitializationCell->True,
  AspectRatioFixed->True]
}, Closed]],

Cell[CellGroupData[{

Cell["\<\
as intersection of a torus
Illustration of slicing a torus to form Cassinian Oval\
\>", "Section"],

Cell[CellGroupData[{

Cell["using PlotRange to cut the torus", "Subsection"],

Cell["\<\
Cassinian ovals are the intersection of a torus and a plane in \
certain position.

Let b-a be the inner radius of a torus whose generating circle has radius a.
Cassinian oval is the intersection of a plane parallel to the torus' axis and \
a distant from it. If b == 2 a, then it's Lemniscate of Bernoulli.\
\>", \
"Text",
  Evaluatable->False,
  AspectRatioFixed->True],

Cell["change b and a for different Cassinian oval.", "Text"],

Cell[CellGroupData[{

Cell[BoxData[
    \(\(With[{a = 1, b = 2.3}, 
        Show[Graphics3D[Torus[b, a, 20, 20]], 
          PlotRange \[Rule] {{\(-1\), 1}\ \((a + b)\), {\(-a\), 
                a + b}, {\(-1\), 1}\ a}, 
          ViewPoint \[Rule] {\(-0.007\), \(-2.017\), 2.717}]];\)\)], "Input",
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Now we wish to add the an outline to the intersection.
The formula of torus with generating circle radius a and center to axis b is
(Sqrt[x^2 + y^2] -b)^2 + z^2 - a^2 == 0
we want to find the intersection of this torus with a plane x=a

so, solve the system of eq. by subsitute x=a,\
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